On the maximal number of real embeddings of spatial minimally rigid graphs

The number of embeddings of minimally rigid graphs in $\mathbb{R}^D$ is (by definition) finite, modulo rigid transformations, for every generic choice of edge lengths. Even though various approaches have been proposed to compute it, the gap between upper and lower bounds is still enormous. Specific values and its asymptotic behavior are major and fascinating open problems in rigidity theory. Our work considers the maximal number of real embeddings of minimally rigid graphs in $\mathbb{R}^3$. We modify a commonly used parametric semi-algebraic formulation that exploits the Cayley-Menger determinant to minimize the {\em a priori} number of complex embeddings, where the parameters correspond to edge lengths. To cope with the huge dimension of the parameter space and find specializations of the parameters that maximize the number of real embeddings, we introduce a method based on coupler curves that makes the sampling feasible for spatial minimally rigid graphs. Our methodology results in the first full classification of the number of real embeddings of graphs with 7 vertices in $\mathbb{R}^3$, which was the smallest open case. Building on this and certain 8-vertex graphs, we improve the previously known general lower bound on the maximum number of real embeddings in $\mathbb{R}^3$

Exact results for the star lattice chiral spin liquid

We examine the star lattice Kitaev model whose ground state is a a chiral spin liquid. We fermionize the model such that the fermionic vacua are toric code states on an effective Kagome lattice. This implies that the Abelian phase of the system is inherited from the fermionic vacua and that time reversal symmetry is spontaneously broken at the level of the vacuum. In terms of these fermions we derive the Bloch-matrix Hamiltonians for the vortex free sector and its time reversed counterpart and illuminate the relationships between the sectors. The phase diagram for the model is shown to be a sphere in the space of coupling parameters around the triangles of the lattices. The abelian phase lies inside the sphere and the critical boundary between topologically distinct Abelian and non-Abelian phases lies on the surface. Outside the sphere the system is generically gapped except in the planes where the coupling parameters are zero. These cases correspond to bipartite lattice structures and the dispersion relations are similar to that of the original Kitaev honeycomb model. In a further analysis we demonstrate the three-fold non-Abelian groundstate degeneracy on a torus by explicit calculation.Comment: 7 pages, 8 figure

On the asymptotic and practical complexity of solving bivariate systems over the reals

This paper is concerned with exact real solving of well-constrained, bivariate polynomial systems. The main problem is to isolate all common real roots in rational rectangles, and to determine their intersection multiplicities. We present three algorithms and analyze their asymptotic bit complexity, obtaining a bound of \sOB(N^{14}) for the purely projection-based method, and \sOB(N^{12}) for two subresultant-based methods: this notation ignores polylogarithmic factors, where $N$ bounds the degree and the bitsize of the polynomials. The previous record bound was \sOB(N^{14}). Our main tool is signed subresultant sequences. We exploit recent advances on the complexity of univariate root isolation, and extend them to sign evaluation of bivariate polynomials over two algebraic numbers, and real root counting for polynomials over an extension field. Our algorithms apply to the problem of simultaneous inequalities; they also compute the topology of real plane algebraic curves in \sOB(N^{12}), whereas the previous bound was \sOB(N^{14}). All algorithms have been implemented in MAPLE, in conjunction with numeric filtering. We compare them against FGB/RS, system solvers from SYNAPS, and MAPLE libraries INSULATE and TOP, which compute curve topology. Our software is among the most robust, and its runtimes are comparable, or within a small constant factor, with respect to the C/C++ libraries. Key words: real solving, polynomial systems, complexity, MAPLE softwareComment: 17 pages, 4 algorithms, 1 table, and 1 figure with 2 sub-figure

Monomial Bases and Polynomial System Solving (Extended Abstract)

This paper addresses the problem of efficient construction of monomial bases for the coordinate rings of zero-dimensional varieties. Existing approaches rely on Gröbner bases methods -- in contrast, we make use of recent developments in sparse elimination techniques which allow us to strongly exploit the structural sparseness of the problem at hand. This is done by establishing certain properties of a matrix formula for the sparse resultant of the given polynomial system. We use this matrix construction to give a simpler proof of the result of Pedersen and Sturmfels [22] for constructing monomial bases. The monomial bases so obtained enable the efficient generation of multiplication maps in coordinate rings and provide a method for computing the common roots of a generic system of polynomial equations with complexity singly exponential in the number of variables and polynomial in the number of roots. We describe the implementations based ..

Predicates for the Planar Additively Weighted Voronoi Diagram

We consider the geometric predicates involved in an incremental algorithm for computing the additively weighted Voronoi diagram in the plane. These predicates correspond to certain algebraic operations, or subpredicates, whose efficient implementation calls for studying various algebraic tools. Our effort is to minimize the algebraic degree of the predicates, thus optimizing the required precision to perform exact arithmetic. We may also try to minimize the number of arithmetic operations; this twofold optimization corresponds to reducing bit complexity. The proposed algorithms are based on Sturm sequences of univariate polynomials and make use of geometric invariants to simplify calculations. Multivariate resultants are also used for a deeper understanding of the predicates and provide an alternative approach to evaluation. We expect that our techniques are sufficiently powerful and general to be applied to a number of analogous geometric problems on curved objects

High-Dimensional Approximate r-Nets

The construction of r-nets offers a powerful tool in computational and metric geometry. We focus on high-dimensional spaces and present a new randomized algorithm which efficiently computes approximate r-nets with respect to Euclidean distance. For any fixed ϵ> 0 , the approximation factor is 1 + ϵ and the complexity is polynomial in the dimension and subquadratic in the number of points; the algorithm succeeds with high probability. Specifically, we improve upon the best previously known (LSH-based) construction of Eppstein et al. (Approximate greedy clustering and distance selection for graph metrics, 2015. CoRR arxiv: abs/1507.01555) in terms of complexity, by reducing the dependence on ϵ, provided that ϵ is sufficiently small. Moreover, our method does not require LSH but follows Valiant’s (J ACM 62(2):13, 2015. https://doi.org/10.1145/2728167) approach in designing a sequence of reductions of our problem to other problems in different spaces, under Euclidean distance or inner product, for which r-nets are computed efficiently and the error can be controlled. Our result immediately implies efficient solutions to a number of geometric problems in high dimension, such as finding the (1 + ϵ) -approximate k-th nearest neighbor distance in time subquadratic in the size of the input. © 2020, Springer Science+Business Media, LLC, part of Springer Nature